What legal measures do real persons/families/groups have to impose "prior restraint" on aspects of a film?

A person's life isn't copyrighted (in the US at least). As long as the person you're making the film about is deceased, you can do what you like... for the most part. Here's an article by Mark Fowler, a legal specialist in this sort of case, (it's about writing but the laws are the same) from a legal writing blog and includes quotes from two major decisions on defamation of the deceased:

• The dead have no cause of action for defamation under the common law, and neither do their survivors, unless the words independently reflect upon and defame the survivors.
  -Judge Robert Sack

• There is no liability for defamation of the dead, either to the estate of the deceased or to the deceased's descendants or relatives.
  -Rodney Smolla

So, as you see here, filmmakers can generally do what they like as long as it doesn't also impact someone who's still alive.

Remember that their documents (books, papers, speeches, music, films, etc.) are copyrighted and a filmmaker can not use them without permission of the person who holds that copyright (usually a spouse or child). It is possible that the producer can make a claim of "fair use" but a film is often very expensive to make and if they lose, they may have to pay damages and the film may never be able to be released.

This was most clearly visible in the recent film Selma. This article from Vox explains why they believe the film opted not to use any of MLK's actual speeches in the film.

If Paramount had distributed a version of Selma that relied on fair use to justify using some King clips, and a court later ruled these uses were not actually fair, it would have been a financial disaster. Paramount could have been on the hook for big damages and could even have been forced to cancel showings of the movie and destroy its inventory of Selma DVDs. And the King estate is famously litigious, having sued both USA Today and CBS for quoting his "I Have a Dream" speech without permission. So rather than take that kind of gamble, studios almost always insist that works be licensed, even if a plausible fair use argument exists.